Solving parametric PDE problems with artificial neural networks

TL;DR

This paper introduces a neural network-based approach to efficiently solve parametric PDE problems, capturing variability with few features and avoiding the curse of dimensionality in uncertainty modeling.

Contribution

The paper proposes a novel neural network method to parameterize PDE solutions based on input coefficients, justified by neural networks performing PDE solution evolution.

Findings

Effective in engineering and physics PDE examples

Avoids curse of dimensionality in uncertainty modeling

Demonstrates simplicity and high accuracy

Abstract

The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from a PDE can be captured by a few features on the space of the coefficient fields. Based on such an observation, we propose using a neural-network (NN) based method to parameterize the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural-network can be justified by viewing the neural-network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.